Uniqueness

Let's explain Uniqueness using examples.

$$\frac{dx}{dt}=ax(t-1)$$ - what is the state of the system?

For $$x(0)$$, we need $$x(-1)$$. But for $$x(0.1)$$, we need $$x(-0.9)$$, and so on...

So $$x_{t}(\theta)=x(t+\theta), \theta \in [-1,0]$$ is the information needed for unambiguous evolution - state of the system

Example: Let $$\Phi_{t}(x_{0})$$ be a state at time $$t$$ given the initial condition $$x_{0}$$.

If $$y=\Phi_{t}(x_{0})$$ and $$z=\Phi_{t'}(y)$$ then $$z=\Phi_{t'+t}(x_{0})$$.

Fundamentally

$$\Phi_{t'+t}=\Phi_{t'}\circ\Phi_{t}$$

Thus from the above we have $$\Phi_{0} = id \,$$ i.e. the identity $$\Phi_{-t} =\Phi_{t}^{-1} $$ i.e. the inverse

Hence the above describe a semigroup.

So the state of the system can be a function. Defining the state of the system may be easy at times and may take a long time.